The Embedding Function

The embedding functional (EB[ ( pB]), which is the key element of the orbital-free embedding formalism reviewed in this chapter, can be seen according to Eq. 39 as a contraction of the functional 5s, , ..., f, 4>nb in which Levy... 

Note that we have imitated Johnson s notation in which the quantity / is a constant factor that appears in the determination of the density p, and the parameters ro, and o are other parameters that are determined through the fitting procedure. Once the density function and the pair potentiai have been specified, fhe determination of the embedding function is complete in principle. In this case, because of the choice of functions, the relevant equations can be solved analytically with the result that the embedding energy is given by... 

The basic strategy used in deriving this form for the embedding function is the recognition that the density function p(r) may be inverted to yield... 

Fig. 4.9. Various representations of the embedding energy for Cu (adapted from Johnson (1988)). The solid curve is the embedding fnnction for Cu derived from UBER as described in text, the dashed curve is a fit for the embedding function and the dotted curve corresponds to an embedding function of the Finnis-Sinclair form in which F(p) a...

This result illustrates the amendment to the vacancy formation energy due to the embedding term. For Johnson s simple analytic model discussed in chap. 4, recall that the embedding function is given by... 

Furthermore, the embedding function and the pair terms are empirically fit to materials properties, and the pair-additive electron contributions <]) are either taken from atomic electron densities or also fit as empirical functions. 

There are two ways of constructing the embedding function F(p). One way is to describe it by an analytical function (or cubic spline) with adjustable parameters. Another way is to postulate an equation of state of the ground-state structure. 

The modified Rose s equation of state (EOS) [86,87], that describes the sublimation energy as a function of lattice constant, (a), is used to determine the embedding function... 

The embedding function F,(/o) is universal , i.e., it does not depend upon the source which creates the electron density. The atomic densities are usually calculated from the tabulated Hartree-Fock wavefunctions [142] and for transition metals taken as a linear combination of the density created by the valence s- and d-electrons, i.e.,... 

The embedding function F(p) is as mentioned supposedly universal but has been determined for various metals [177] and also given an analytical form (see, e.g., Truong et al. [209]). The effective charge Z(R) is expressed as [177]... 

Here, Fi is the embedding function and represents the energy necessary to embed an atom i in the electronic density pnj, at the site where this atom is located. ph,i is calculated as the superposition of the individual electronic densities Piiry) ... 

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